AND ENUMERATION OF POLYEDRA. 55 



faces triangular y has not more than J(q — 1) triedral sum- 

 mits, when q i* odd, nor more than i(q — 2), when q is even, 



9. Let us now consider the n-edron P, which has an 

 (n — l)-gonal base A, all its summits triedral, and more 

 than two — suppose, for example, four — triangular faces. 



It is necessary to fix, first, the position of the triangles. 

 The first being numbered one, the other three will be the 

 (c+2)% the (c+c,+3)"S and the (f+e,+e,+4)% of the n 



faces abed so that e faces lie between the first and 



second, c, between the second and third, c, between the third 

 and fourth, and (n — e — e, — e, — 5) between the fourth and 

 first triangle ; where 



n — e — c, — e^ — 570, or 

 e4-€,+c,Zn— 5. 



Taking, for example, w=l5, we may examine the case of 



e=\, c,=2, e,=6, n — e — c, — e^ — 5=1. 



The triangles are then a, c, f, m, which determine the triplets 



nab, bed, efg, Imn, 

 in addition to the fourteen 



Kab, Abe, Amn, Ana. 



Of the 39(=3w^-6) edges, there are 14 just written in 

 duads of consecutive letters ab, be, &c., of which eight are 

 repeated in the four preceding triplets, viz., na, ab, be, &c. 

 There remain six more duads of consecutive letters to be 

 employed, of which no two can form a triplet ; for the middle 

 letter of such a triplet would denote a fifth triangle. But 

 there are eight triplets yet to be determined to complete the 

 26 (=2/1 — 4) of the system. There must therefore be two 

 triplets constructed which contain no duad of consecutive 

 letters. These denote what I shall call ridge-summits. 



If aye be one of them, the edge ay is the intersection of 

 two non-consecutive faces a and y ; and it is evident, that if 

 we proceed along this edge in either direction, we shall find 



